Assignment 2: Key Sampling Distributions in Statistical Inference

1 Abstract

This paper summarizes the normal, t, chi-square, and F distributions, their assumptions, and their connections in statistical inference. These distributions arise from sampling and form the basis for inference on means, variances, and variance ratios.

2 Normal Distribution

• Description: The normal distribution is a symmetric distribution characterized by a mean and variance and is commonly used to model random variation in data. When data are drawn from a normal population, the sample mean is normally distributed.

• Assumptions: The normal distribution assumes that observations are independent and come from a population with a symmetric, bell-shaped distribution characterized by a finite mean and variance.

• Common use: Modeling natural variation; inference for means when variance is known.

x <- seq(-4, 4, length = 1000)
y <- dnorm(x, mean = 0, sd = 1)

plot(x, y, type = "l", lwd = 2, col = "blue",
     main = "Normal Distribution N(0,1)",
     xlab = "x", ylab = "Density")

3 Student’s t Distribution

• Description: The t distribution is bell-shaped like the normal but has heavier tails, especially for small degrees of freedom.

• Assumptions: The t distribution assumes that the sample is drawn independently from a normal population and that the population variance is unknown and estimated using the sample variance.

• Common use: Inference for the population mean when variance is unknown.

x <- seq(-4, 4, length = 1000)
y <- dt(x, df = 5)

plot(x, y, type = "l", lwd = 2, col = "red",
     main = "Student's t Distribution (df = 5)",
     xlab = "x", ylab = "Density")

4 Chi-Square Distribution

• Description: The chi-square distribution is defined on \([0,\infty)\), is usually right-skewed, and depends on degrees of freedom.

• Assumptions: The chi-square distribution assumes independent observations from a normal population and arises from the sum of squared standardized normal variables, making it suitable for inference about variance.

• Common use: Inference about population variance; goodness-of-fit and independence tests.

x <- seq(0, 20, length = 1000)
y <- dchisq(x, df = 5)

plot(x, y, type = "l", lwd = 2, col = "darkgreen",
     main = "Chi-Square Distribution (df = 5)",
     xlab = "x", ylab = "Density")

5 F Distribution

• Description: The F distribution is defined on \([0,\infty)\), is right-skewed, and depends on two degrees of freedom.

• Assumptions: The F distribution assumes two independent samples drawn from normal populations and is based on the ratio of their sample variances.

• Common use: Comparing variances; analysis of variance (ANOVA).

x <- seq(0, 5, length = 1000)
y <- df(x, df1 = 5, df2 = 10)

plot(x, y, type = "l", lwd = 2, col = "purple",
     main = "F Distribution (df1 = 5, df2 = 10)",
     xlab = "x", ylab = "Density")

6 Connections Among the Distributions

The normal distribution provides the starting point for the t, chi-square, and F distributions. The t distribution is built by combining a normal random variable with information from a chi-square distribution, while the chi-square distribution itself comes from adding together squared standard normal variables. The F distribution is formed by taking the ratio of two independent chi-square distributions. Because of these relationships, all four distributions naturally arise in sampling problems and are widely used in hypothesis testing and confidence interval construction.